let X be non empty set ; :: thesis:
let F be Functional_Sequence of X,ExtREAL; :: thesis:
let n be Nat; :: thesis:
defpred S₁[ Nat] means (Partial_Sums F) . $1 is V124();
assume A1: for m being Nat holds F . m is V124() ; :: thesis:
A2: for k being Nat st S₁[k] holds
S₁[k + 1]

let k be Nat; :: thesis:
assume A3: S₁[k] ; :: thesis:
A4: F . (k + 1) is V124() by A1;
(Partial_Sums F) . (k + 1) = ((Partial_Sums F) . k) + (F . (k + 1)) by Def4;
hence S₁[k + 1] by A3, A4, Th3; :: thesis:

end;